Problem 43 - College Algebra with Modeling and Visualization

Question

Apply the distributive property. $$5 x(x-5)$$

Step-by-Step Solution

Answer

The expression \(5x(x-5)\) simplifies to \(5x^2 - 25x\).

Step 1: Identify the Expression

The expression given is \(5x(x-5)\). We need to apply the distributive property to simplify this expression.

Step 2: Apply the Distributive Property

Use the distributive property, \(a(b + c) = ab + ac\), to distribute \(5x\) over each term in the parentheses: \(5x(x) + 5x(-5)\).

Step 3: Simplify Each Term

Calculate each term from Step 2. First, \(5x \times x\) results in \(5x^2\). Then, \(5x \times -5\) results in \(-25x\).

Step 4: Combine the Terms

Combine the terms obtained: \(5x^2 - 25x\). This is the simplified expression.

Key Concepts

Polynomial Simplification

Polynomial simplification involves reducing expressions to a simpler or more concise form without changing their value. In essence, you are breaking down complex polynomials into their simplest parts. Simplifying polynomials makes them easier to work with and solve. One common method is using the Distributive Property, which helps in expanding and then combining like terms.

For example, consider the polynomial expression given in our exercise:

• Starting with the expression: \(5x(x - 5)\), use the Distributive Property to rewrite it by multiplying the outside term \(5x\) by each term inside the parenthesis.
• This means you compute \(5x \times x\) and \(5x \times -5\).
• The results \(5x^2\) and \(-25x\) are then combined to give a simplified polynomial \(5x^2 - 25x\).

By simplifying expressions in this way, you not only make them easier to understand but also prepare them for solving equations or graphing.

Algebraic Expressions

An algebraic expression is a mathematical phrase that contains numbers, variables, and operation symbols. It could be something as simple as \(x + 2\) or more complex, like \(5x(x - 5)\).

In algebra, expressions represent relationships between quantities and are foundational for solving equations and understanding functions. Here are a few key insights about algebraic expressions:

• Variables: Symbols like \(x\), \(y\), or any letter represent unknown values and can change.
• Coefficients: Numbers like \(5\) in \(5x\), which multiply the variable, are called coefficients.
• Operators: These are symbols such as \(+\), \(-\), and \(\times\) that tell us how to combine numbers and variables.
• Terms: Parts of the expression separated by plus or minus signs. In \(5x^2 - 25x\), \(5x^2\) and \(-25x\) are terms.

Understanding these components is crucial for manipulating and solving algebraic expressions effectively.

Step-by-step Solutions

Using step-by-step solutions is an effective way to tackle algebra problems, especially when working with polynomial expressions. Breaking down a problem into manageable parts allows you to focus on one task at a time, leading to a complete understanding of the process.

Let's see how this works by examining the distributive law in our case:

• Step 1: Identify the expression you need to work on. Here it is \(5x(x - 5)\).
• Step 2: Apply the distributive property to multiply the single term by each term inside the parenthesis.
• Step 3: Simplify each resulting term. Calculate \(5x^2\) from \(5x \times x\) and \(-25x\) from \(5x \times -5\).
• Step 4: Combine these terms to reach the final simplified expression: \(5x^2 - 25x\).

Following these steps helps in organizing your thought process and ensures that no part of the expression is left out or miscalculated. Step-by-step methods build greater confidence in problem-solving, as you learn to tackle each component individually and effectively.